phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
subgacent1E : 'C_(S \| to)[a] = 'Fix_(S \| to)[a].	Proof. by rewrite gacent1E setIA (setIidPl sSR). Qed.	Lemma	subgacent1E	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacent1E", "sSR", "setIA", "setIidPl", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gact1 : {in D, forall a, to 1 a = 1}.	Proof. by move=> a Da; rewrite /= -actmE ?morph1. Qed.	Lemma	gact1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actmE", "morph1", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gactV : {in D, forall a, {in R, {morph to^~ a : x / x^-1}}}.	Proof. by move=> a Da /= x Rx; move; rewrite -!actmE ?morphV. Qed.	Lemma	gactV	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actmE", "morphV", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gactX : {in D, forall a n, {in R, {morph to^~ a : x / x ^+ n}}}.	Proof. by move=> a Da /= n x Rx; rewrite -!actmE // morphX. Qed.	Lemma	gactX	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actmE", "morphX", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gactJ : {in D, forall a, {in R &, {morph to^~ a : x y / x ^ y}}}.	Proof. by move=> a Da /= x Rx y Ry; rewrite -!actmE // morphJ. Qed.	Lemma	gactJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actmE", "morphJ", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gactR : {in D, forall a, {in R &, {morph to^~ a : x y / [~ x, y]}}}.	Proof. by move=> a Da /= x Rx y Ry; rewrite -!actmE // morphR. Qed.	Lemma	gactR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actmE", "morphR", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gact_stable : {acts D, on R \| to}.	Proof. apply: acts_act; apply/subsetP=> a Da; rewrite !inE Da. apply/subsetP=> x; rewrite inE; apply: contraLR => R'xa. by rewrite -(actKin to Da x) gact_out ?groupV. Qed.	Lemma	gact_stable	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actKin", "acts_act", "apply", "gact_out", "groupV", "inE", "on", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_gacent A : group_set 'C_(\|to)(A).	Proof. apply/group_setP; split=> [\|x y]. by rewrite !inE group1; apply/subsetP=> a /setIP[Da _]; rewrite inE gact1. case/setIP=> Rx /afixP cAx /setIP[Ry /afixP cAy]. rewrite inE groupM //; apply/afixP=> a Aa. by rewrite gactM ?cAx ?cAy //; case/setIP: Aa. Qed.	Lemma	group_set_gacent	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "afixP", "apply", "gact1", "gactM", "group1", "groupM", "group_set", "group_setP", "inE", "setIP", "split", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent_group A	:= Group (group_set_gacent A).	Canonical	gacent_group	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "group_set_gacent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent1 : 'C_(\|to)(1) = R.	Proof. by rewrite /gacent (setIidPr (sub1G _)) afix1 setIT. Qed.	Lemma	gacent1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afix1", "gacent", "setIT", "setIidPr", "sub1G", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent_gen A : A \subset D -> 'C_(\|to)(<<A>>) = 'C_(\|to)(A).	Proof. by move=> sAD; rewrite /gacent ![D :&: _](setIidPr _) ?gen_subG ?afix_gen_in. Qed.	Lemma	gacent_gen	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afix_gen_in", "gacent", "gen_subG", "sAD", "setIidPr", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentD1 A : 'C_(\|to)(A^#) = 'C_(\|to)(A).	Proof. rewrite -gacentIdom -gacent_gen ?subsetIl // setIDA genD1 ?group1 //. by rewrite gacent_gen ?subsetIl // gacentIdom. Qed.	Lemma	gacentD1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacentIdom", "gacent_gen", "genD1", "group1", "setIDA", "subsetIl", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent_cycle a : a \in D -> 'C_(\|to)(<[a]>) = 'C_(\|to)[a].	Proof. by move=> Da; rewrite gacent_gen ?sub1set. Qed.	Lemma	gacent_cycle	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "gacent_gen", "sub1set", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentY A B : A \subset D -> B \subset D -> 'C_(\|to)(A <*> B) = 'C_(\|to)(A) :&: 'C_(\|to)(B).	Proof. by move=> sAD sBD; rewrite gacent_gen ?gacentU // subUset sAD. Qed.	Lemma	gacentY	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacentU", "gacent_gen", "sAD", "subUset", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentM G H : G \subset D -> H \subset D -> 'C_(\|to)(G * H) = 'C_(\|to)(G) :&: 'C_(\|to)(H).	Proof. by move=> sGD sHB; rewrite -gacent_gen ?mul_subG // genM_join gacentY. Qed.	Lemma	gacentM	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacentY", "gacent_gen", "genM_join", "mul_subG", "sGD", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1 : 'C(1 \| to) = D.	Proof. by apply/setP=> x; rewrite ?(inE, sub1set) andb_idr //; move/gact1=> ->. Qed.	Lemma	astab1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "gact1", "inE", "setP", "sub1set", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_range : 'C(R \| to) = 'C(setT \| to).	Proof. apply/eqP; rewrite eqEsubset andbC astabS ?subsetT //=. apply/subsetP=> a cRa; have Da := astab_dom cRa; rewrite !inE Da. apply/subsetP=> x; rewrite -(setUCr R) !inE. by case/orP=> ?; [rewrite (astab_act cRa) \| rewrite gact_out]. Qed.	Lemma	astab_range	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "apply", "astabS", "astab_act", "astab_dom", "eqEsubset", "gact_out", "inE", "setT", "setUCr", "subsetP", "subsetT", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentC A S : A \subset D -> S \subset R -> (S \subset 'C_(\|to)(A)) = (A \subset 'C(S \| to)).	Proof. by move=> sAD sSR; rewrite subsetI sSR astabCin // (setIidPr sAD). Qed.	Lemma	gacentC	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabCin", "sAD", "sSR", "setIidPr", "subsetI", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab_gen S : S \subset R -> 'C(<<S>> \| to) = 'C(S \| to).	Proof. move=> sSR; apply/setP=> a; case Da: (a \in D); last by rewrite !inE Da. by rewrite -!sub1set -!gacentC ?sub1set ?gen_subG. Qed.	Lemma	astab_gen	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "apply", "gacentC", "gen_subG", "inE", "last", "sSR", "setP", "sub1set", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabM M N : M \subset R -> N \subset R -> 'C(M * N \| to) = 'C(M \| to) :&: 'C(N \| to).	Proof. move=> sMR sNR; rewrite -astabU -astab_gen ?mul_subG // genM_join. by rewrite astab_gen // subUset sMR. Qed.	Lemma	astabM	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabU", "astab_gen", "genM_join", "mul_subG", "subUset", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs1 : 'N(1 \| to) = D.	Proof. by rewrite astabs_set1 astab1. Qed.	Lemma	astabs1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astab1", "astabs_set1", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabs_range : 'N(R \| to) = D.	Proof. apply/setIidPl; apply/subsetP=> a Da; rewrite inE. by apply/subsetP=> x Rx; rewrite inE gact_stable. Qed.	Lemma	astabs_range	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "apply", "gact_stable", "inE", "setIidPl", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsD1 S : 'N(S^# \| to) = 'N(S \| to).	Proof. case S1: (1 \in S); last first. by rewrite (setDidPl _) // disjoint_sym disjoints_subset sub1set inE S1. apply/eqP; rewrite eqEsubset andbC -{1}astabsIdom -{1}astabs1 setIC astabsD /=. by rewrite -{2}(setD1K S1) -astabsIdom -{1}astabs1 astabsU. Qed.	Lemma	astabsD1	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "S1", "apply", "astabs1", "astabsD", "astabsIdom", "astabsU", "disjoint_sym", "disjoints_subset", "eqEsubset", "inE", "last", "setD1K", "setDidPl", "setIC", "sub1set", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacts_range A : A \subset D -> {acts A, on group R \| to}.	Proof. by move=> sAD; split; rewrite ?astabs_range. Qed.	Lemma	gacts_range	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabs_range", "group", "on", "sAD", "split", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_subnorm_gacent A : A \subset D -> [acts 'N_D(A), on 'C_(\| to)(A) \| to].	Proof. move=> sAD; rewrite gacentE // actsI ?astabs_range ?subsetIl //. by rewrite -{2}(setIidPr sAD) acts_subnorm_fix. Qed.	Lemma	acts_subnorm_gacent	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actsI", "acts_subnorm_fix", "astabs_range", "gacentE", "on", "sAD", "setIidPr", "subsetIl", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_subnorm_subgacent A B S : A \subset D -> [acts B, on S \| to] -> [acts 'N_B(A), on 'C_(S \| to)(A) \| to].	Proof. move=> sAD actsB; rewrite actsI //; first by rewrite subIset ?actsB. by rewrite (subset_trans _ (acts_subnorm_gacent sAD)) ?setSI ?(acts_dom actsB). Qed.	Lemma	acts_subnorm_subgacent	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actsI", "acts_dom", "acts_subnorm_gacent", "on", "sAD", "setSI", "subIset", "subset_trans", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_gen A S : S \subset R -> [acts A, on S \| to] -> [acts A, on <<S>> \| to].	Proof. move=> sSR actsA; apply: {A}subset_trans actsA _. apply/subsetP=> a nSa; have Da := astabs_dom nSa; rewrite !inE Da. apply: subset_trans (_ : <<S>> \subset actm to a @*^-1 <<S>>) _. rewrite gen_subG subsetI sSR; apply/subsetP=> x Sx. by rewrite inE /= actmE ?mem_gen // astabs_act. by apply/subsetP=> x /[!inE...	Lemma	acts_gen	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actm", "actmE", "apply", "astabs_act", "astabs_dom", "gen_subG", "inE", "mem_gen", "on", "sSR", "subsetI", "subsetP", "subset_trans", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_joing A M N : M \subset R -> N \subset R -> [acts A, on M \| to] -> [acts A, on N \| to] -> [acts A, on M <*> N \| to].	Proof. by move=> sMR sNR nMA nNA; rewrite acts_gen ?actsU // subUset sMR. Qed.	Lemma	acts_joing	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actsU", "acts_gen", "on", "subUset", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_actm a : 'injm (actm to a).	Proof. apply/injmP=> x y Rx Ry; rewrite /= /actm; case: ifP => Da //. exact: act_inj. Qed.	Lemma	injm_actm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "act_inj", "actm", "apply", "injmP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_actm a : actm to a @* R = R.	Proof. apply/eqP; rewrite eqEcard (card_injm (injm_actm a)) // leqnn andbT. apply/subsetP=> _ /morphimP[x Rx _ ->] /=. by rewrite /actm; case: ifP => // Da; rewrite gact_stable. Qed.	Lemma	im_actm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actm", "apply", "card_injm", "eqEcard", "gact_stable", "injm_actm", "leqnn", "morphimP", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_char G M : G \subset D -> M \char R -> [acts G, on M \| to].	Proof. move=> sGD /charP[sMR charM]. apply/subsetP=> a Ga; have Da := subsetP sGD a Ga; rewrite !inE Da. apply/subsetP=> x Mx; have Rx := subsetP sMR x Mx. by rewrite inE -(charM _ (injm_actm a) (im_actm a)) -actmE // mem_morphim. Qed.	Lemma	acts_char	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actmE", "apply", "char", "charM", "charP", "im_actm", "inE", "injm_actm", "mem_morphim", "on", "sGD", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacts_char G M : G \subset D -> M \char R -> {acts G, on group M \| to}.	(* TODO: investigate why rewrite does not match in the same order *) Proof. by move=> sGD charM; split; rewrite ?acts_char// char_sub. Qed.	Lemma	gacts_char	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_char", "char", "charM", "char_sub", "group", "on", "sGD", "split", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ract_is_groupAction : is_groupAction R (to \ sAD).	Proof. by move=> a Aa /=; rewrite ractpermE actperm_Aut ?(subsetP sAD). Qed.	Lemma	ract_is_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm_Aut", "is_groupAction", "ractpermE", "sAD", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ract_groupAction	:= GroupAction ract_is_groupAction.	Canonical	ract_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "ract_is_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent_ract B : 'C_(\|ract_groupAction)(B) = 'C_(\|to)(A :&: B).	Proof. by rewrite /gacent afix_ract setIA (setIidPr sAD). Qed.	Lemma	gacent_ract	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afix_ract", "gacent", "ract_groupAction", "sAD", "setIA", "setIidPr", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actby_is_groupAction : is_groupAction G <[nGAg]>.	Proof. move=> a Aa; rewrite /= inE; apply/andP; split. apply/subsetP=> x; apply: contraR => Gx. by rewrite actpermE /= /actby (negbTE Gx). apply/morphicP=> x y Gx Gy; rewrite !actpermE /= /actby Aa groupM ?Gx ?Gy //=. by case nGAg; move/acts_dom; do 2!move/subsetP=> ?; rewrite gactM; auto. Qed.	Lemma	actby_is_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby", "actpermE", "acts_dom", "apply", "gactM", "groupM", "inE", "is_groupAction", "morphicP", "split", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actby_groupAction	:= GroupAction actby_is_groupAction.	Canonical	actby_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby_is_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent_actby B : 'C_(\|actby_groupAction)(B) = 'C_(G \| to)(A :&: B).	Proof. rewrite /gacent afix_actby !setIA setIid setIUr setICr set0U. by have [nAG sGR] := nGAg; rewrite (setIidPr (acts_dom nAG)) (setIidPl sGR). Qed.	Lemma	gacent_actby	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby_groupAction", "acts_dom", "afix_actby", "gacent", "set0U", "setIA", "setICr", "setIUr", "setIid", "setIidPl", "setIidPr", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_qact_dom_norm : {acts qact_dom to H, on 'N(H) \| to}.	Proof. move=> a HDa /= x; rewrite {2}(('N(H) =P to^~ a @^-1: 'N(H)) _) ?inE {x}//. rewrite eqEcard (card_preimset _ (act_inj _ _)) leqnn andbT. apply/subsetP=> x Nx; rewrite inE; move/(astabs_act (H :* x)): HDa. rewrite mem_rcosets mulSGid ?normG // Nx => /rcosetsP[y Ny defHy]. suffices: to x a \in H :* y by apply: sub...	Lemma	acts_qact_dom_norm	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act_inj", "apply", "astabs_act", "card_preimset", "eqEcard", "imset_f", "inE", "leqnn", "mem_rcosets", "mulSGid", "mul_subG", "normG", "on", "qact_dom", "rcoset_refl", "rcosetsP", "sub1set", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qact_is_groupAction : is_groupAction (R / H) (to / H).	Proof. move=> a HDa /=; have Da := astabs_dom HDa. rewrite inE; apply/andP; split. apply/subsetP=> Hx /=; case: (cosetP Hx) => x Nx ->{Hx}. apply: contraR => R'Hx; rewrite actpermE qactE // gact_out //. by apply: contra R'Hx; apply: mem_morphim. apply/morphicP=> Hx Hy; rewrite !actpermE. case/morphimP=> x Nx Gx -...	Lemma	qact_is_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actpermE", "acts_qact_dom_norm", "apply", "astabs_dom", "cosetP", "gactM", "gact_out", "groupM", "inE", "is_groupAction", "mem_morphim", "morphM", "morphicP", "morphimP", "qactE", "split", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_groupAction	:= GroupAction qact_is_groupAction.	Canonical	quotient_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "qact_is_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qact_domE : H \subset R -> qact_dom to H = 'N(H \| to).	Proof. move=> sHR; apply/setP=> a; apply/idP/idP=> nHa; have Da := astabs_dom nHa. rewrite !inE Da; apply/subsetP=> x Hx; rewrite inE -(rcoset1 H). have /rcosetsP[y Ny defHy]: to^~ a @: H \in rcosets H 'N(H). by rewrite (astabs_act _ nHa); apply/rcosetsP; exists 1; rewrite ?mulg1. by rewrite (rcoset_eqP (_ : ...	Lemma	qact_domE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actKVin", "act_inj", "apply", "astabs_act", "astabs_dom", "card_imset", "card_rcoset", "eqEcard", "gact1", "gactJ", "gactM", "gactV", "gact_out", "groupMl", "groupMr", "groupV", "imsetP", "inE", "last", "leqnn", "memJ_norm", "mem_rcoset", "mem_setact", "mulg1",...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modact_is_groupAction : is_groupAction 'C_(\|to)(H) (to %% H).	Proof. move=> Ha /morphimP[a Na Da ->]; have NDa: a \in 'N_D(H) by apply/setIP. rewrite inE; apply/andP; split. apply/subsetP=> x; rewrite 2!inE andbC actpermE /= modactEcond //. by apply: contraR; case: ifP => // E Rx; rewrite gact_out. apply/morphicP=> x y /setIP[Rx cHx] /setIP[Ry cHy]. rewrite /= !actpermE /= !m...	Lemma	modact_is_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "actpermE", "apply", "gactM", "gact_out", "groupM", "inE", "is_groupAction", "modactE", "modactEcond", "morphicP", "morphimP", "setIP", "split", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mod_groupAction	:= GroupAction modact_is_groupAction.	Canonical	mod_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "modact_is_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modgactE x a : H \subset 'C(R \| to) -> a \in 'N_D(H) -> (to %% H)%act x (coset H a) = to x a.	Proof. move=> cRH NDa /=; have [Da Na] := setIP NDa. have [Rx \| notRx] := boolP (x \in R). by rewrite modactE //; apply/afixP=> b /setIP[_ /(subsetP cRH)/astab_act->]. rewrite gact_out //= /modact; case: ifP => // _; rewrite gact_out //. suffices: a \in D :&: coset H a by case/mem_repr/setIP. by rewrite inE Da val_co...	Lemma	modgactE	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "act", "afixP", "apply", "astab_act", "coset", "gact_out", "inE", "mem_repr", "modact", "modactE", "rcoset_refl", "setIP", "subsetP", "to", "val_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent_mod G M : H \subset 'C(M \| to) -> G \subset 'N(H) -> 'C_(M \| mod_groupAction)(G / H) = 'C_(M \| to)(G).	Proof. move=> cMH nHG; rewrite -gacentIdom gacentE ?subsetIl // setICA. have sHD: H \subset D by rewrite (subset_trans cMH) ?subsetIl. rewrite -quotientGI // afix_mod ?setIS // setICA -gacentIim (setIC R) -setIA. rewrite -gacentE ?subsetIl // gacentIdom setICA (setIidPr _) //. by rewrite gacentC // ?(subset_trans cMH) ...	Lemma	gacent_mod	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afix_mod", "astabS", "gacentC", "gacentE", "gacentIdom", "gacentIim", "mod_groupAction", "nHG", "quotientGI", "sHD", "setIA", "setIC", "setICA", "setIS", "setIidPr", "subsetIl", "subset_trans", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_irr_mod G M : H \subset 'C(M \| to) -> G \subset 'N(H) -> acts_irreducibly G M to -> acts_irreducibly (G / H) M mod_groupAction.	Proof. move=> cMH nHG /mingroupP[/andP[ntM nMG] minM]. apply/mingroupP; rewrite ntM astabs_mod ?quotientS //; split=> // L modL ntL. have cLH: H \subset 'C(L \| to) by rewrite (subset_trans cMH) ?astabS //. apply: minM => //; case/andP: modL => ->; rewrite astabs_mod ?quotientSGK //. by rewrite (subset_trans cLH) ?astab...	Lemma	acts_irr_mod	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_irreducibly", "apply", "astabS", "astab_sub", "astabs_mod", "mingroupP", "mod_groupAction", "nHG", "nMG", "quotientS", "quotientSGK", "split", "subset_trans", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
modact_coset_astab x a : a \in D -> (to %% 'C(R \| to))%act x (coset _ a) = to x a.	Proof. move=> Da; apply: modgactE => {x}//. rewrite !inE Da; apply/subsetP=> _ /imsetP[c Cc ->]. have Dc := astab_dom Cc; rewrite !inE groupJ //. apply/subsetP=> x Rx; rewrite inE conjgE !actMin ?groupM ?groupV //. by rewrite (astab_act Cc) ?actKVin // gact_stable ?groupV. Qed.	Lemma	modact_coset_astab	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Da", "act", "actKVin", "actMin", "apply", "astab_act", "astab_dom", "conjgE", "coset", "gact_stable", "groupJ", "groupM", "groupV", "imsetP", "inE", "modgactE", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
acts_irr_mod_astab G M : acts_irreducibly G M to -> acts_irreducibly (G / 'C_G(M \| to)) M (mod_groupAction _).	Proof. move=> irrG; have /andP[_ nMG] := mingroupp irrG. apply: acts_irr_mod irrG; first exact: subsetIr. by rewrite normsI ?normG // (subset_trans nMG) // astab_norm. Qed.	Lemma	acts_irr_mod_astab	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "acts_irr_mod", "acts_irreducibly", "apply", "astab_norm", "irrG", "mingroupp", "mod_groupAction", "nMG", "normG", "normsI", "subsetIr", "subset_trans", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_is_groupAction : is_groupAction R (comp_action to f).	Proof. move=> a /morphpreP[Ba Dfa]; apply: etrans (actperm_Aut to Dfa). by congr (_ \in Aut R); apply/permP=> x; rewrite !actpermE. Qed.	Lemma	comp_is_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Aut", "actpermE", "actperm_Aut", "apply", "comp_action", "is_groupAction", "morphpreP", "permP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_groupAction	:= GroupAction comp_is_groupAction.	Canonical	comp_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "comp_is_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacent_comp U : 'C_(\|comp_groupAction)(U) = 'C_(\|to)(f @* U).	Proof. rewrite /gacent afix_comp ?subIset ?subxx //. by rewrite -(setIC U) (setIC D) morphim_setIpre. Qed.	Lemma	gacent_comp	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "afix_comp", "comp_groupAction", "gacent", "morphim_setIpre", "setIC", "subIset", "subxx", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( \| to ) ( A )"	:= (gacent_group to A) : Group_scope.	Notation	''C_' ( \| to ) ( A )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacent_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( G \| to ) ( A )"	:= (setI_group G 'C_(\|to)(A)) : Group_scope.	Notation	''C_' ( G \| to ) ( A )	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setI_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( \| to ) [ a ]"	:= (gacent_group to [set a%g]) : Group_scope.	Notation	''C_' ( \| to ) [ a ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gacent_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''C_' ( G \| to ) [ a ]"	:= (setI_group G 'C_(\|to)[a]) : Group_scope.	Notation	''C_' ( G \| to ) [ a ]	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "setI_group", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to \ sAD"	:= (ract_groupAction to sAD) : groupAction_scope.	Notation	to \ sAD	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "ract_groupAction", "sAD", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<[ nGA ] >"	:= (actby_groupAction nGA) : groupAction_scope.	Notation	<[ nGA ] >	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actby_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to / H"	:= (quotient_groupAction to H) : groupAction_scope.	Notation	to / H	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "quotient_groupAction", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to %% H"	:= (mod_groupAction to H) : groupAction_scope.	Notation	to %% H	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "mod_groupAction", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"to \o f"	:= (comp_groupAction to f) : groupAction_scope.	Notation	to \o f	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "comp_groupAction", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(actsDR : {acts D1, on R \| to1}) (injh : {in R &, injective h}).		Hypotheses	actsDR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "on", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
defD2 : f @* D1 = D2.		Hypothesis	defD2	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(sSR : S \subset R) (sAD1 : A \subset D1).		Hypotheses	sSR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hfJ : {in S & D1, morph_act to1 to2 h f}.		Hypothesis	hfJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "morph_act", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_astabs : f @* 'N(S \| to1) = 'N(h @: S \| to2).	Proof. apply/setP=> fx; apply/morphimP/idP=> [[x D1x nSx ->] \| nSx]. rewrite 2!inE -{1}defD2 mem_morphim //=; apply/subsetP=> _ /imsetP[u Su ->]. by rewrite inE -hfJ ?imset_f // (astabs_act _ nSx). have [\|x D1x _ def_fx] := morphimP (_ : fx \in f @* D1). by rewrite defD2 (astabs_dom nSx). exists x => //; rewrite ...	Lemma	morph_astabs	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actsDR", "apply", "astabs_act", "astabs_dom", "defD2", "hfJ", "imsetP", "imset_f", "inE", "mem_morphim", "morphimP", "sSR", "setP", "subsetP", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_astab : f @* 'C(S \| to1) = 'C(h @: S \| to2).	Proof. apply/setP=> fx; apply/morphimP/idP=> [[x D1x cSx ->] \| cSx]. rewrite 2!inE -{1}defD2 mem_morphim //=; apply/subsetP=> _ /imsetP[u Su ->]. by rewrite inE -hfJ // (astab_act cSx). have [\|x D1x _ def_fx] := morphimP (_ : fx \in f @* D1). by rewrite defD2 (astab_dom cSx). exists x => //; rewrite !inE D1x; app...	Lemma	morph_astab	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actsDR", "apply", "astab_act", "astab_dom", "defD2", "hfJ", "imsetP", "imset_f", "inE", "inj_in_eq", "mem_morphim", "morphimP", "sSR", "setP", "subsetP", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_afix : h @: 'Fix_(S \| to1)(A) = 'Fix_(h @: S \| to2)(f @* A).	Proof. apply/setP=> hu; apply/imsetP/setIP=> [[u /setIP[Su cAu] ->]\|]. split; first by rewrite imset_f. by apply/afixP=> _ /morphimP[x D1x Ax ->]; rewrite -hfJ ?(afixP cAu). case=> /imsetP[u Su ->] /afixP c_hu_fA; exists u; rewrite // inE Su. apply/afixP=> x Ax; have Dx := subsetP sAD1 x Ax. by apply: injh; rewrite...	Lemma	morph_afix	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Dx", "actsDR", "afixP", "apply", "hfJ", "imsetP", "imset_f", "inE", "mem_morphim", "morphimP", "sSR", "setIP", "setP", "split", "subsetP", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(iso_h : isom R1 R2 h) (iso_f : isom D1 D2 f).		Hypotheses	iso_h	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "R1", "R2", "isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hfJ : {in R1 & D1, morph_act to1 to2 h f}.		Hypothesis	hfJ	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "R1", "morph_act", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_gastabs S : S \subset R1 -> f @* 'N(S \| to1) = 'N(h @* S \| to2).	Proof. have [[_ defD2] [injh _]] := (isomP iso_f, isomP iso_h). move=> sSR1; rewrite (morphimEsub _ sSR1). apply: (morph_astabs (gact_stable to1) (injmP injh)) => // u x. by move/(subsetP sSR1); apply: hfJ. Qed.	Lemma	morph_gastabs	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "R1", "apply", "defD2", "gact_stable", "hfJ", "injmP", "iso_h", "isomP", "morph_astabs", "morphimEsub", "subsetP", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_gastab S : S \subset R1 -> f @* 'C(S \| to1) = 'C(h @* S \| to2).	Proof. have [[_ defD2] [injh _]] := (isomP iso_f, isomP iso_h). move=> sSR1; rewrite (morphimEsub _ sSR1). apply: (morph_astab (gact_stable to1) (injmP injh)) => // u x. by move/(subsetP sSR1); apply: hfJ. Qed.	Lemma	morph_gastab	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "R1", "apply", "defD2", "gact_stable", "hfJ", "injmP", "iso_h", "isomP", "morph_astab", "morphimEsub", "subsetP", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_gacent A : A \subset D1 -> h @* 'C_(\|to1)(A) = 'C_(\|to2)(f @* A).	Proof. have [[_ defD2] [injh defR2]] := (isomP iso_f, isomP iso_h). move=> sAD1; rewrite !gacentE //; first by rewrite -defD2 morphimS. rewrite morphimEsub ?subsetIl // -{1}defR2 morphimEdom. exact: (morph_afix (gact_stable to1) (injmP injh)). Qed.	Lemma	morph_gacent	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "defD2", "gacentE", "gact_stable", "injmP", "iso_h", "isomP", "morph_afix", "morphimEdom", "morphimEsub", "morphimS", "subsetIl", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_gact_irr A M : A \subset D1 -> M \subset R1 -> acts_irreducibly (f @* A) (h @* M) to2 = acts_irreducibly A M to1.	Proof. move=> sAD1 sMR1. have [[injf defD2] [injh defR2]] := (isomP iso_f, isomP iso_h). have h_eq1 := morphim_injm_eq1 injh. apply/mingroupP/mingroupP=> [] [/andP[ntM actAM] minM]. split=> [\|U]; first by rewrite -h_eq1 // ntM -(injmSK injf) ?morph_gastabs. case/andP=> ntU acts_fAU sUM; have sUR1 := subset_trans sU...	Lemma	morph_gact_irr	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "R1", "acts_irreducibly", "apply", "defD2", "injf", "injmSK", "injm_morphim_inj", "iso_h", "isomP", "last", "mingroupP", "morph_gastabs", "morphimS", "morphim_injm_eq1", "morphpreK", "split", "sub_morphpre_injm", "subsetIl", "subset_trans", "to1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulgr_action	:= TotalAction (@mulg1 gT) (@mulgA gT).	Definition	mulgr_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "TotalAction", "gT", "mulg1", "mulgA" ]	action!).	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_action	:= TotalAction (@conjg1 gT) (@conjgM gT).	Canonical	conjg_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "TotalAction", "conjg1", "conjgM", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_is_groupAction : is_groupAction setT conjg_action.	Proof. move=> a _; rewrite inE; apply/andP; split; first by apply/subsetP=> x /[1!inE]. by apply/morphicP=> x y _ _; rewrite !actpermE /= conjMg. Qed.	Lemma	conjg_is_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actpermE", "apply", "conjMg", "conjg_action", "inE", "is_groupAction", "morphicP", "setT", "split", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjg_groupAction	:= GroupAction conjg_is_groupAction.	Canonical	conjg_groupAction	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "conjg_is_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_is_action : is_action setT (@rcoset gT).	Proof. by apply: is_total_action => [A\|A x y]; rewrite !rcosetE (mulg1, rcosetM). Qed.	Lemma	rcoset_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "gT", "is_action", "is_total_action", "mulg1", "rcoset", "rcosetE", "rcosetM", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_action	:= Action rcoset_is_action.	Canonical	rcoset_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "rcoset_is_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjsg_action	:= TotalAction (@conjsg1 gT) (@conjsgM gT).	Canonical	conjsg_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "TotalAction", "conjsg1", "conjsgM", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjG_is_action : is_action setT (@conjG_group gT).	Proof. apply: is_total_action => [G \| G x y]; apply: val_inj; rewrite /= ?act1 //. exact: actM. Qed.	Lemma	conjG_is_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act1", "actM", "apply", "conjG_group", "gT", "is_action", "is_total_action", "setT", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
conjG_action	:= Action conjG_is_action.	Definition	conjG_action	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "conjG_is_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'R"	:= (@mulgr_action _) : action_scope.	Notation	'R	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "mulgr_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'Rs"	:= (@rcoset_action _) : action_scope.	Notation	'Rs	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "rcoset_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'J"	:= (@conjg_action _) : action_scope.	Notation	'J	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "conjg_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'J"	:= (@conjg_groupAction _) : groupAction_scope.	Notation	'J	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "conjg_groupAction" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'Js"	:= (@conjsg_action _) : action_scope.	Notation	'Js	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "conjsg_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'JG"	:= (@conjG_action _) : action_scope.	Notation	'JG	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "conjG_action" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'Q"	:= ('J / _)%act : action_scope.	Notation	'Q	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "act" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'Q"	:= ('J / _)%gact : groupAction_scope.	Notation	'Q	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "gact" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
orbitR G x : orbit 'R G x = x *: G.	Proof. by rewrite -lcosetE. Qed.	Lemma	orbitR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "lcosetE", "orbit" ]	Various identities for actions on groups.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astab1R x : 'C[x \| 'R] = 1.	Proof. apply/trivgP/subsetP=> y cxy. by rewrite -(mulKg x y) [x * y](astab1P cxy) mulVg set11. Qed.	Lemma	astab1R	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astab1P", "mulKg", "mulVg", "set11", "subsetP", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabR G : 'C(G \| 'R) = 1.	Proof. apply/trivgP/subsetP=> x cGx. by rewrite -(mul1g x) [1 * x](astabP cGx) group1. Qed.	Lemma	astabR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "astabP", "group1", "mul1g", "subsetP", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsR G : 'N(G \| 'R) = G.	Proof. apply/setP=> x; rewrite !inE -setactVin ?inE //=. by rewrite -groupV -{1 3}(mulg1 G) rcoset_sym -sub1set -mulGS -!rcosetE. Qed.	Lemma	astabsR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "apply", "groupV", "inE", "mulGS", "mulg1", "rcosetE", "rcoset_sym", "setP", "setactVin", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
atransR G : [transitive G, on G \| 'R].	Proof. by rewrite /atrans -{1}(mul1g G) -orbitR imset_f. Qed.	Lemma	atransR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "atrans", "imset_f", "mul1g", "on", "orbitR" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
faithfulR G : [faithful G, on G \| 'R].	Proof. by rewrite /faithful astabR subsetIr. Qed.	Lemma	faithfulR	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "astabR", "faithful", "on", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Cayley_repr G	:= actperm <[atrans_acts (atransR G)]>.	Definition	Cayley_repr	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "actperm", "atransR", "atrans_acts" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Cayley_isom G : isom G (Cayley_repr G @* G) (Cayley_repr G).	Proof. exact: faithful_isom (faithfulR G). Qed.	Theorem	Cayley_isom	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Cayley_repr", "faithfulR", "faithful_isom", "isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Cayley_isog G : G \isog Cayley_repr G @* G.	Proof. exact: isom_isog (Cayley_isom G). Qed.	Theorem	Cayley_isog	finite_group	finite_group/action.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]	[ "Cayley_isom", "Cayley_repr", "isog", "isom_isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgacent1E : 'C_(S | to)[a] = 'Fix_(S | to)[a].

Proof. by rewrite gacent1E setIA (setIidPl sSR). Qed.

Lemma

subgacent1E

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacent1E", "sSR", "setIA", "setIidPl", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gact1 : {in D, forall a, to 1 a = 1}.

Proof. by move=> a Da; rewrite /= -actmE ?morph1. Qed.

Lemma

gact1

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actmE", "morph1", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gactV : {in D, forall a, {in R, {morph to^~ a : x / x^-1}}}.

Proof. by move=> a Da /= x Rx; move; rewrite -!actmE ?morphV. Qed.

Lemma

gactV

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actmE", "morphV", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gactX : {in D, forall a n, {in R, {morph to^~ a : x / x ^+ n}}}.

Proof. by move=> a Da /= n x Rx; rewrite -!actmE // morphX. Qed.

Lemma

gactX

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actmE", "morphX", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gactJ : {in D, forall a, {in R &, {morph to^~ a : x y / x ^ y}}}.

Proof. by move=> a Da /= x Rx y Ry; rewrite -!actmE // morphJ. Qed.

Lemma

gactJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actmE", "morphJ", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gactR : {in D, forall a, {in R &, {morph to^~ a : x y / [~ x, y]}}}.

Proof. by move=> a Da /= x Rx y Ry; rewrite -!actmE // morphR. Qed.

Lemma

gactR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actmE", "morphR", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gact_stable : {acts D, on R | to}.

Proof. apply: acts_act; apply/subsetP=> a Da; rewrite !inE Da. apply/subsetP=> x; rewrite inE; apply: contraLR => R'xa. by rewrite -(actKin to Da x) gact_out ?groupV. Qed.

Lemma

gact_stable

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actKin", "acts_act", "apply", "gact_out", "groupV", "inE", "on", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_gacent A : group_set 'C_(|to)(A).

Proof. apply/group_setP; split=> [|x y]. by rewrite !inE group1; apply/subsetP=> a /setIP[Da _]; rewrite inE gact1. case/setIP=> Rx /afixP cAx /setIP[Ry /afixP cAy]. rewrite inE groupM //; apply/afixP=> a Aa. by rewrite gactM ?cAx ?cAy //; case/setIP: Aa. Qed.

Lemma

group_set_gacent

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "afixP", "apply", "gact1", "gactM", "group1", "groupM", "group_set", "group_setP", "inE", "setIP", "split", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent_group A

:= Group (group_set_gacent A).

Canonical

gacent_group

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "group_set_gacent" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent1 : 'C_(|to)(1) = R.

Proof. by rewrite /gacent (setIidPr (sub1G _)) afix1 setIT. Qed.

Lemma

gacent1

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afix1", "gacent", "setIT", "setIidPr", "sub1G", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent_gen A : A \subset D -> 'C_(|to)(<<A>>) = 'C_(|to)(A).

Proof. by move=> sAD; rewrite /gacent ![D :&: _](setIidPr _) ?gen_subG ?afix_gen_in. Qed.

Lemma

gacent_gen

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afix_gen_in", "gacent", "gen_subG", "sAD", "setIidPr", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentD1 A : 'C_(|to)(A^#) = 'C_(|to)(A).

Proof. rewrite -gacentIdom -gacent_gen ?subsetIl // setIDA genD1 ?group1 //. by rewrite gacent_gen ?subsetIl // gacentIdom. Qed.

Lemma

gacentD1

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacentIdom", "gacent_gen", "genD1", "group1", "setIDA", "subsetIl", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent_cycle a : a \in D -> 'C_(|to)(<[a]>) = 'C_(|to)[a].

Proof. by move=> Da; rewrite gacent_gen ?sub1set. Qed.

Lemma

gacent_cycle

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "gacent_gen", "sub1set", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentY A B : A \subset D -> B \subset D -> 'C_(|to)(A <*> B) = 'C_(|to)(A) :&: 'C_(|to)(B).

Proof. by move=> sAD sBD; rewrite gacent_gen ?gacentU // subUset sAD. Qed.

Lemma

gacentY

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacentU", "gacent_gen", "sAD", "subUset", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentM G H : G \subset D -> H \subset D -> 'C_(|to)(G * H) = 'C_(|to)(G) :&: 'C_(|to)(H).

Proof. by move=> sGD sHB; rewrite -gacent_gen ?mul_subG // genM_join gacentY. Qed.

Lemma

gacentM

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacentY", "gacent_gen", "genM_join", "mul_subG", "sGD", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1 : 'C(1 | to) = D.

Proof. by apply/setP=> x; rewrite ?(inE, sub1set) andb_idr //; move/gact1=> ->. Qed.

Lemma

astab1

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "gact1", "inE", "setP", "sub1set", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_range : 'C(R | to) = 'C(setT | to).

Proof. apply/eqP; rewrite eqEsubset andbC astabS ?subsetT //=. apply/subsetP=> a cRa; have Da := astab_dom cRa; rewrite !inE Da. apply/subsetP=> x; rewrite -(setUCr R) !inE. by case/orP=> ?; [rewrite (astab_act cRa) | rewrite gact_out]. Qed.

Lemma

astab_range

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "apply", "astabS", "astab_act", "astab_dom", "eqEsubset", "gact_out", "inE", "setT", "setUCr", "subsetP", "subsetT", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentC A S : A \subset D -> S \subset R -> (S \subset 'C_(|to)(A)) = (A \subset 'C(S | to)).

Proof. by move=> sAD sSR; rewrite subsetI sSR astabCin // (setIidPr sAD). Qed.

Lemma

gacentC

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabCin", "sAD", "sSR", "setIidPr", "subsetI", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab_gen S : S \subset R -> 'C(<<S>> | to) = 'C(S | to).

Proof. move=> sSR; apply/setP=> a; case Da: (a \in D); last by rewrite !inE Da. by rewrite -!sub1set -!gacentC ?sub1set ?gen_subG. Qed.

Lemma

astab_gen

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "apply", "gacentC", "gen_subG", "inE", "last", "sSR", "setP", "sub1set", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabM M N : M \subset R -> N \subset R -> 'C(M * N | to) = 'C(M | to) :&: 'C(N | to).

Proof. move=> sMR sNR; rewrite -astabU -astab_gen ?mul_subG // genM_join. by rewrite astab_gen // subUset sMR. Qed.

Lemma

astabM

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabU", "astab_gen", "genM_join", "mul_subG", "subUset", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs1 : 'N(1 | to) = D.

Proof. by rewrite astabs_set1 astab1. Qed.

Lemma

astabs1

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astab1", "astabs_set1", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabs_range : 'N(R | to) = D.

Proof. apply/setIidPl; apply/subsetP=> a Da; rewrite inE. by apply/subsetP=> x Rx; rewrite inE gact_stable. Qed.

Lemma

astabs_range

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "apply", "gact_stable", "inE", "setIidPl", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsD1 S : 'N(S^# | to) = 'N(S | to).

Proof. case S1: (1 \in S); last first. by rewrite (setDidPl _) // disjoint_sym disjoints_subset sub1set inE S1. apply/eqP; rewrite eqEsubset andbC -{1}astabsIdom -{1}astabs1 setIC astabsD /=. by rewrite -{2}(setD1K S1) -astabsIdom -{1}astabs1 astabsU. Qed.

Lemma

astabsD1

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "S1", "apply", "astabs1", "astabsD", "astabsIdom", "astabsU", "disjoint_sym", "disjoints_subset", "eqEsubset", "inE", "last", "setD1K", "setDidPl", "setIC", "sub1set", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacts_range A : A \subset D -> {acts A, on group R | to}.

Proof. by move=> sAD; split; rewrite ?astabs_range. Qed.

Lemma

gacts_range

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabs_range", "group", "on", "sAD", "split", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_subnorm_gacent A : A \subset D -> [acts 'N_D(A), on 'C_(| to)(A) | to].

Proof. move=> sAD; rewrite gacentE // actsI ?astabs_range ?subsetIl //. by rewrite -{2}(setIidPr sAD) acts_subnorm_fix. Qed.

Lemma

acts_subnorm_gacent

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actsI", "acts_subnorm_fix", "astabs_range", "gacentE", "on", "sAD", "setIidPr", "subsetIl", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_subnorm_subgacent A B S : A \subset D -> [acts B, on S | to] -> [acts 'N_B(A), on 'C_(S | to)(A) | to].

Proof. move=> sAD actsB; rewrite actsI //; first by rewrite subIset ?actsB. by rewrite (subset_trans _ (acts_subnorm_gacent sAD)) ?setSI ?(acts_dom actsB). Qed.

Lemma

acts_subnorm_subgacent

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actsI", "acts_dom", "acts_subnorm_gacent", "on", "sAD", "setSI", "subIset", "subset_trans", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_gen A S : S \subset R -> [acts A, on S | to] -> [acts A, on <<S>> | to].

Proof. move=> sSR actsA; apply: {A}subset_trans actsA _. apply/subsetP=> a nSa; have Da := astabs_dom nSa; rewrite !inE Da. apply: subset_trans (_ : <<S>> \subset actm to a @*^-1 <<S>>) _. rewrite gen_subG subsetI sSR; apply/subsetP=> x Sx. by rewrite inE /= actmE ?mem_gen // astabs_act. by apply/subsetP=> x /[!inE...

Lemma

acts_gen

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actm", "actmE", "apply", "astabs_act", "astabs_dom", "gen_subG", "inE", "mem_gen", "on", "sSR", "subsetI", "subsetP", "subset_trans", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_joing A M N : M \subset R -> N \subset R -> [acts A, on M | to] -> [acts A, on N | to] -> [acts A, on M <*> N | to].

Proof. by move=> sMR sNR nMA nNA; rewrite acts_gen ?actsU // subUset sMR. Qed.

Lemma

acts_joing

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actsU", "acts_gen", "on", "subUset", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_actm a : 'injm (actm to a).

Proof. apply/injmP=> x y Rx Ry; rewrite /= /actm; case: ifP => Da //. exact: act_inj. Qed.

Lemma

injm_actm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "act_inj", "actm", "apply", "injmP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_actm a : actm to a @* R = R.

Proof. apply/eqP; rewrite eqEcard (card_injm (injm_actm a)) // leqnn andbT. apply/subsetP=> _ /morphimP[x Rx _ ->] /=. by rewrite /actm; case: ifP => // Da; rewrite gact_stable. Qed.

Lemma

im_actm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actm", "apply", "card_injm", "eqEcard", "gact_stable", "injm_actm", "leqnn", "morphimP", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_char G M : G \subset D -> M \char R -> [acts G, on M | to].

Proof. move=> sGD /charP[sMR charM]. apply/subsetP=> a Ga; have Da := subsetP sGD a Ga; rewrite !inE Da. apply/subsetP=> x Mx; have Rx := subsetP sMR x Mx. by rewrite inE -(charM _ (injm_actm a) (im_actm a)) -actmE // mem_morphim. Qed.

Lemma

acts_char

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actmE", "apply", "char", "charM", "charP", "im_actm", "inE", "injm_actm", "mem_morphim", "on", "sGD", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacts_char G M : G \subset D -> M \char R -> {acts G, on group M | to}.

(* TODO: investigate why rewrite does not match in the same order *) Proof. by move=> sGD charM; split; rewrite ?acts_char// char_sub. Qed.

Lemma

gacts_char

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_char", "char", "charM", "char_sub", "group", "on", "sGD", "split", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ract_is_groupAction : is_groupAction R (to \ sAD).

Proof. by move=> a Aa /=; rewrite ractpermE actperm_Aut ?(subsetP sAD). Qed.

Lemma

ract_is_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm_Aut", "is_groupAction", "ractpermE", "sAD", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ract_groupAction

:= GroupAction ract_is_groupAction.

Canonical

ract_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "ract_is_groupAction" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent_ract B : 'C_(|ract_groupAction)(B) = 'C_(|to)(A :&: B).

Proof. by rewrite /gacent afix_ract setIA (setIidPr sAD). Qed.

Lemma

gacent_ract

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afix_ract", "gacent", "ract_groupAction", "sAD", "setIA", "setIidPr", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actby_is_groupAction : is_groupAction G <[nGAg]>.

Proof. move=> a Aa; rewrite /= inE; apply/andP; split. apply/subsetP=> x; apply: contraR => Gx. by rewrite actpermE /= /actby (negbTE Gx). apply/morphicP=> x y Gx Gy; rewrite !actpermE /= /actby Aa groupM ?Gx ?Gy //=. by case nGAg; move/acts_dom; do 2!move/subsetP=> ?; rewrite gactM; auto. Qed.

Lemma

actby_is_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby", "actpermE", "acts_dom", "apply", "gactM", "groupM", "inE", "is_groupAction", "morphicP", "split", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actby_groupAction

:= GroupAction actby_is_groupAction.

Canonical

actby_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby_is_groupAction" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent_actby B : 'C_(|actby_groupAction)(B) = 'C_(G | to)(A :&: B).

Proof. rewrite /gacent afix_actby !setIA setIid setIUr setICr set0U. by have [nAG sGR] := nGAg; rewrite (setIidPr (acts_dom nAG)) (setIidPl sGR). Qed.

Lemma

gacent_actby

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby_groupAction", "acts_dom", "afix_actby", "gacent", "set0U", "setIA", "setICr", "setIUr", "setIid", "setIidPl", "setIidPr", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_qact_dom_norm : {acts qact_dom to H, on 'N(H) | to}.

Proof. move=> a HDa /= x; rewrite {2}(('N(H) =P to^~ a @^-1: 'N(H)) _) ?inE {x}//. rewrite eqEcard (card_preimset _ (act_inj _ _)) leqnn andbT. apply/subsetP=> x Nx; rewrite inE; move/(astabs_act (H :* x)): HDa. rewrite mem_rcosets mulSGid ?normG // Nx => /rcosetsP[y Ny defHy]. suffices: to x a \in H :* y by apply: sub...

Lemma

acts_qact_dom_norm

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act_inj", "apply", "astabs_act", "card_preimset", "eqEcard", "imset_f", "inE", "leqnn", "mem_rcosets", "mulSGid", "mul_subG", "normG", "on", "qact_dom", "rcoset_refl", "rcosetsP", "sub1set", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qact_is_groupAction : is_groupAction (R / H) (to / H).

Proof. move=> a HDa /=; have Da := astabs_dom HDa. rewrite inE; apply/andP; split. apply/subsetP=> Hx /=; case: (cosetP Hx) => x Nx ->{Hx}. apply: contraR => R'Hx; rewrite actpermE qactE // gact_out //. by apply: contra R'Hx; apply: mem_morphim. apply/morphicP=> Hx Hy; rewrite !actpermE. case/morphimP=> x Nx Gx -...

Lemma

qact_is_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actpermE", "acts_qact_dom_norm", "apply", "astabs_dom", "cosetP", "gactM", "gact_out", "groupM", "inE", "is_groupAction", "mem_morphim", "morphM", "morphicP", "morphimP", "qactE", "split", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_groupAction

:= GroupAction qact_is_groupAction.

Canonical

quotient_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "qact_is_groupAction" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qact_domE : H \subset R -> qact_dom to H = 'N(H | to).

Proof. move=> sHR; apply/setP=> a; apply/idP/idP=> nHa; have Da := astabs_dom nHa. rewrite !inE Da; apply/subsetP=> x Hx; rewrite inE -(rcoset1 H). have /rcosetsP[y Ny defHy]: to^~ a @: H \in rcosets H 'N(H). by rewrite (astabs_act _ nHa); apply/rcosetsP; exists 1; rewrite ?mulg1. by rewrite (rcoset_eqP (_ : ...

Lemma

qact_domE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actKVin", "act_inj", "apply", "astabs_act", "astabs_dom", "card_imset", "card_rcoset", "eqEcard", "gact1", "gactJ", "gactM", "gactV", "gact_out", "groupMl", "groupMr", "groupV", "imsetP", "inE", "last", "leqnn", "memJ_norm", "mem_rcoset", "mem_setact", "mulg1",...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modact_is_groupAction : is_groupAction 'C_(|to)(H) (to %% H).

Proof. move=> Ha /morphimP[a Na Da ->]; have NDa: a \in 'N_D(H) by apply/setIP. rewrite inE; apply/andP; split. apply/subsetP=> x; rewrite 2!inE andbC actpermE /= modactEcond //. by apply: contraR; case: ifP => // E Rx; rewrite gact_out. apply/morphicP=> x y /setIP[Rx cHx] /setIP[Ry cHy]. rewrite /= !actpermE /= !m...

Lemma

modact_is_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "actpermE", "apply", "gactM", "gact_out", "groupM", "inE", "is_groupAction", "modactE", "modactEcond", "morphicP", "morphimP", "setIP", "split", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mod_groupAction

:= GroupAction modact_is_groupAction.

Canonical

mod_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "modact_is_groupAction" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modgactE x a : H \subset 'C(R | to) -> a \in 'N_D(H) -> (to %% H)%act x (coset H a) = to x a.

Proof. move=> cRH NDa /=; have [Da Na] := setIP NDa. have [Rx | notRx] := boolP (x \in R). by rewrite modactE //; apply/afixP=> b /setIP[_ /(subsetP cRH)/astab_act->]. rewrite gact_out //= /modact; case: ifP => // _; rewrite gact_out //. suffices: a \in D :&: coset H a by case/mem_repr/setIP. by rewrite inE Da val_co...

Lemma

modgactE

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "act", "afixP", "apply", "astab_act", "coset", "gact_out", "inE", "mem_repr", "modact", "modactE", "rcoset_refl", "setIP", "subsetP", "to", "val_coset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent_mod G M : H \subset 'C(M | to) -> G \subset 'N(H) -> 'C_(M | mod_groupAction)(G / H) = 'C_(M | to)(G).

Proof. move=> cMH nHG; rewrite -gacentIdom gacentE ?subsetIl // setICA. have sHD: H \subset D by rewrite (subset_trans cMH) ?subsetIl. rewrite -quotientGI // afix_mod ?setIS // setICA -gacentIim (setIC R) -setIA. rewrite -gacentE ?subsetIl // gacentIdom setICA (setIidPr _) //. by rewrite gacentC // ?(subset_trans cMH) ...

Lemma

gacent_mod

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afix_mod", "astabS", "gacentC", "gacentE", "gacentIdom", "gacentIim", "mod_groupAction", "nHG", "quotientGI", "sHD", "setIA", "setIC", "setICA", "setIS", "setIidPr", "subsetIl", "subset_trans", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_irr_mod G M : H \subset 'C(M | to) -> G \subset 'N(H) -> acts_irreducibly G M to -> acts_irreducibly (G / H) M mod_groupAction.

Proof. move=> cMH nHG /mingroupP[/andP[ntM nMG] minM]. apply/mingroupP; rewrite ntM astabs_mod ?quotientS //; split=> // L modL ntL. have cLH: H \subset 'C(L | to) by rewrite (subset_trans cMH) ?astabS //. apply: minM => //; case/andP: modL => ->; rewrite astabs_mod ?quotientSGK //. by rewrite (subset_trans cLH) ?astab...

Lemma

acts_irr_mod

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_irreducibly", "apply", "astabS", "astab_sub", "astabs_mod", "mingroupP", "mod_groupAction", "nHG", "nMG", "quotientS", "quotientSGK", "split", "subset_trans", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

modact_coset_astab x a : a \in D -> (to %% 'C(R | to))%act x (coset _ a) = to x a.

Proof. move=> Da; apply: modgactE => {x}//. rewrite !inE Da; apply/subsetP=> _ /imsetP[c Cc ->]. have Dc := astab_dom Cc; rewrite !inE groupJ //. apply/subsetP=> x Rx; rewrite inE conjgE !actMin ?groupM ?groupV //. by rewrite (astab_act Cc) ?actKVin // gact_stable ?groupV. Qed.

Lemma

modact_coset_astab

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Da", "act", "actKVin", "actMin", "apply", "astab_act", "astab_dom", "conjgE", "coset", "gact_stable", "groupJ", "groupM", "groupV", "imsetP", "inE", "modgactE", "subsetP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

acts_irr_mod_astab G M : acts_irreducibly G M to -> acts_irreducibly (G / 'C_G(M | to)) M (mod_groupAction _).

Proof. move=> irrG; have /andP[_ nMG] := mingroupp irrG. apply: acts_irr_mod irrG; first exact: subsetIr. by rewrite normsI ?normG // (subset_trans nMG) // astab_norm. Qed.

Lemma

acts_irr_mod_astab

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "acts_irr_mod", "acts_irreducibly", "apply", "astab_norm", "irrG", "mingroupp", "mod_groupAction", "nMG", "normG", "normsI", "subsetIr", "subset_trans", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_is_groupAction : is_groupAction R (comp_action to f).

Proof. move=> a /morphpreP[Ba Dfa]; apply: etrans (actperm_Aut to Dfa). by congr (_ \in Aut R); apply/permP=> x; rewrite !actpermE. Qed.

Lemma

comp_is_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Aut", "actpermE", "actperm_Aut", "apply", "comp_action", "is_groupAction", "morphpreP", "permP", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_groupAction

:= GroupAction comp_is_groupAction.

Canonical

comp_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "comp_is_groupAction" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacent_comp U : 'C_(|comp_groupAction)(U) = 'C_(|to)(f @* U).

Proof. rewrite /gacent afix_comp ?subIset ?subxx //. by rewrite -(setIC U) (setIC D) morphim_setIpre. Qed.

Lemma

gacent_comp

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "afix_comp", "comp_groupAction", "gacent", "morphim_setIpre", "setIC", "subIset", "subxx", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( | to ) ( A )"

:= (gacent_group to A) : Group_scope.

Notation

''C_' ( | to ) ( A )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacent_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( G | to ) ( A )"

:= (setI_group G 'C_(|to)(A)) : Group_scope.

Notation

''C_' ( G | to ) ( A )

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "setI_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( | to ) [ a ]"

:= (gacent_group to [set a%g]) : Group_scope.

Notation

''C_' ( | to ) [ a ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gacent_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''C_' ( G | to ) [ a ]"

:= (setI_group G 'C_(|to)[a]) : Group_scope.

Notation

''C_' ( G | to ) [ a ]

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "setI_group", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to \ sAD"

:= (ract_groupAction to sAD) : groupAction_scope.

Notation

to \ sAD

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "ract_groupAction", "sAD", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<[ nGA ] >"

:= (actby_groupAction nGA) : groupAction_scope.

Notation

<[ nGA ] >

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actby_groupAction" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to / H"

:= (quotient_groupAction to H) : groupAction_scope.

Notation

to / H

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "quotient_groupAction", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to %% H"

:= (mod_groupAction to H) : groupAction_scope.

Notation

to %% H

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "mod_groupAction", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"to \o f"

:= (comp_groupAction to f) : groupAction_scope.

Notation

to \o f

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "comp_groupAction", "to" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(actsDR : {acts D1, on R | to1}) (injh : {in R &, injective h}).

Hypotheses

actsDR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "on", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

defD2 : f @* D1 = D2.

Hypothesis

defD2

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(sSR : S \subset R) (sAD1 : A \subset D1).

Hypotheses

sSR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hfJ : {in S & D1, morph_act to1 to2 h f}.

Hypothesis

hfJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "morph_act", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_astabs : f @* 'N(S | to1) = 'N(h @: S | to2).

Proof. apply/setP=> fx; apply/morphimP/idP=> [[x D1x nSx ->] | nSx]. rewrite 2!inE -{1}defD2 mem_morphim //=; apply/subsetP=> _ /imsetP[u Su ->]. by rewrite inE -hfJ ?imset_f // (astabs_act _ nSx). have [|x D1x _ def_fx] := morphimP (_ : fx \in f @* D1). by rewrite defD2 (astabs_dom nSx). exists x => //; rewrite ...

Lemma

morph_astabs

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actsDR", "apply", "astabs_act", "astabs_dom", "defD2", "hfJ", "imsetP", "imset_f", "inE", "mem_morphim", "morphimP", "sSR", "setP", "subsetP", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_astab : f @* 'C(S | to1) = 'C(h @: S | to2).

Proof. apply/setP=> fx; apply/morphimP/idP=> [[x D1x cSx ->] | cSx]. rewrite 2!inE -{1}defD2 mem_morphim //=; apply/subsetP=> _ /imsetP[u Su ->]. by rewrite inE -hfJ // (astab_act cSx). have [|x D1x _ def_fx] := morphimP (_ : fx \in f @* D1). by rewrite defD2 (astab_dom cSx). exists x => //; rewrite !inE D1x; app...

Lemma

morph_astab

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actsDR", "apply", "astab_act", "astab_dom", "defD2", "hfJ", "imsetP", "imset_f", "inE", "inj_in_eq", "mem_morphim", "morphimP", "sSR", "setP", "subsetP", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_afix : h @: 'Fix_(S | to1)(A) = 'Fix_(h @: S | to2)(f @* A).

Proof. apply/setP=> hu; apply/imsetP/setIP=> [[u /setIP[Su cAu] ->]|]. split; first by rewrite imset_f. by apply/afixP=> _ /morphimP[x D1x Ax ->]; rewrite -hfJ ?(afixP cAu). case=> /imsetP[u Su ->] /afixP c_hu_fA; exists u; rewrite // inE Su. apply/afixP=> x Ax; have Dx := subsetP sAD1 x Ax. by apply: injh; rewrite...

Lemma

morph_afix

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Dx", "actsDR", "afixP", "apply", "hfJ", "imsetP", "imset_f", "inE", "mem_morphim", "morphimP", "sSR", "setIP", "setP", "split", "subsetP", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(iso_h : isom R1 R2 h) (iso_f : isom D1 D2 f).

Hypotheses

iso_h

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "R1", "R2", "isom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hfJ : {in R1 & D1, morph_act to1 to2 h f}.

Hypothesis

hfJ

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "R1", "morph_act", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_gastabs S : S \subset R1 -> f @* 'N(S | to1) = 'N(h @* S | to2).

Proof. have [[_ defD2] [injh _]] := (isomP iso_f, isomP iso_h). move=> sSR1; rewrite (morphimEsub _ sSR1). apply: (morph_astabs (gact_stable to1) (injmP injh)) => // u x. by move/(subsetP sSR1); apply: hfJ. Qed.

Lemma

morph_gastabs

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "R1", "apply", "defD2", "gact_stable", "hfJ", "injmP", "iso_h", "isomP", "morph_astabs", "morphimEsub", "subsetP", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_gastab S : S \subset R1 -> f @* 'C(S | to1) = 'C(h @* S | to2).

Proof. have [[_ defD2] [injh _]] := (isomP iso_f, isomP iso_h). move=> sSR1; rewrite (morphimEsub _ sSR1). apply: (morph_astab (gact_stable to1) (injmP injh)) => // u x. by move/(subsetP sSR1); apply: hfJ. Qed.

Lemma

morph_gastab

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "R1", "apply", "defD2", "gact_stable", "hfJ", "injmP", "iso_h", "isomP", "morph_astab", "morphimEsub", "subsetP", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_gacent A : A \subset D1 -> h @* 'C_(|to1)(A) = 'C_(|to2)(f @* A).

Proof. have [[_ defD2] [injh defR2]] := (isomP iso_f, isomP iso_h). move=> sAD1; rewrite !gacentE //; first by rewrite -defD2 morphimS. rewrite morphimEsub ?subsetIl // -{1}defR2 morphimEdom. exact: (morph_afix (gact_stable to1) (injmP injh)). Qed.

Lemma

morph_gacent

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "defD2", "gacentE", "gact_stable", "injmP", "iso_h", "isomP", "morph_afix", "morphimEdom", "morphimEsub", "morphimS", "subsetIl", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_gact_irr A M : A \subset D1 -> M \subset R1 -> acts_irreducibly (f @* A) (h @* M) to2 = acts_irreducibly A M to1.

Proof. move=> sAD1 sMR1. have [[injf defD2] [injh defR2]] := (isomP iso_f, isomP iso_h). have h_eq1 := morphim_injm_eq1 injh. apply/mingroupP/mingroupP=> [] [/andP[ntM actAM] minM]. split=> [|U]; first by rewrite -h_eq1 // ntM -(injmSK injf) ?morph_gastabs. case/andP=> ntU acts_fAU sUM; have sUR1 := subset_trans sU...

Lemma

morph_gact_irr

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "R1", "acts_irreducibly", "apply", "defD2", "injf", "injmSK", "injm_morphim_inj", "iso_h", "isomP", "last", "mingroupP", "morph_gastabs", "morphimS", "morphim_injm_eq1", "morphpreK", "split", "sub_morphpre_injm", "subsetIl", "subset_trans", "to1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulgr_action

:= TotalAction (@mulg1 gT) (@mulgA gT).

Definition

mulgr_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "TotalAction", "gT", "mulg1", "mulgA" ]

action!).

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg_action

:= TotalAction (@conjg1 gT) (@conjgM gT).

Canonical

conjg_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "TotalAction", "conjg1", "conjgM", "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg_is_groupAction : is_groupAction setT conjg_action.

Proof. move=> a _; rewrite inE; apply/andP; split; first by apply/subsetP=> x /[1!inE]. by apply/morphicP=> x y _ _; rewrite !actpermE /= conjMg. Qed.

Lemma

conjg_is_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actpermE", "apply", "conjMg", "conjg_action", "inE", "is_groupAction", "morphicP", "setT", "split", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjg_groupAction

:= GroupAction conjg_is_groupAction.

Canonical

conjg_groupAction

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "conjg_is_groupAction" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_is_action : is_action setT (@rcoset gT).

Proof. by apply: is_total_action => [A|A x y]; rewrite !rcosetE (mulg1, rcosetM). Qed.

Lemma

rcoset_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "gT", "is_action", "is_total_action", "mulg1", "rcoset", "rcosetE", "rcosetM", "setT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_action

:= Action rcoset_is_action.

Canonical

rcoset_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "rcoset_is_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjsg_action

:= TotalAction (@conjsg1 gT) (@conjsgM gT).

Canonical

conjsg_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "TotalAction", "conjsg1", "conjsgM", "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjG_is_action : is_action setT (@conjG_group gT).

Proof. apply: is_total_action => [G | G x y]; apply: val_inj; rewrite /= ?act1 //. exact: actM. Qed.

Lemma

conjG_is_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act1", "actM", "apply", "conjG_group", "gT", "is_action", "is_total_action", "setT", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

conjG_action

:= Action conjG_is_action.

Definition

conjG_action

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "conjG_is_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'R"

:= (@mulgr_action _) : action_scope.

Notation

'R

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "mulgr_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'Rs"

:= (@rcoset_action _) : action_scope.

Notation

'Rs

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "rcoset_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'J"

:= (@conjg_action _) : action_scope.

Notation

'J

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "conjg_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'J"

:= (@conjg_groupAction _) : groupAction_scope.

Notation

'J

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "conjg_groupAction" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'Js"

:= (@conjsg_action _) : action_scope.

Notation

'Js

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "conjsg_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'JG"

:= (@conjG_action _) : action_scope.

Notation

'JG

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "conjG_action" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'Q"

:= ('J / _)%act : action_scope.

Notation

'Q

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "act" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"'Q"

:= ('J / _)%gact : groupAction_scope.

Notation

'Q

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "gact" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

orbitR G x : orbit 'R G x = x *: G.

Proof. by rewrite -lcosetE. Qed.

Lemma

orbitR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "lcosetE", "orbit" ]

Various identities for actions on groups.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astab1R x : 'C[x | 'R] = 1.

Proof. apply/trivgP/subsetP=> y cxy. by rewrite -(mulKg x y) [x * y](astab1P cxy) mulVg set11. Qed.

Lemma

astab1R

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astab1P", "mulKg", "mulVg", "set11", "subsetP", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabR G : 'C(G | 'R) = 1.

Proof. apply/trivgP/subsetP=> x cGx. by rewrite -(mul1g x) [1 * x](astabP cGx) group1. Qed.

Lemma

astabR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "astabP", "group1", "mul1g", "subsetP", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsR G : 'N(G | 'R) = G.

Proof. apply/setP=> x; rewrite !inE -setactVin ?inE //=. by rewrite -groupV -{1 3}(mulg1 G) rcoset_sym -sub1set -mulGS -!rcosetE. Qed.

Lemma

astabsR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "apply", "groupV", "inE", "mulGS", "mulg1", "rcosetE", "rcoset_sym", "setP", "setactVin", "sub1set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

atransR G : [transitive G, on G | 'R].

Proof. by rewrite /atrans -{1}(mul1g G) -orbitR imset_f. Qed.

Lemma

atransR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "atrans", "imset_f", "mul1g", "on", "orbitR" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

faithfulR G : [faithful G, on G | 'R].

Proof. by rewrite /faithful astabR subsetIr. Qed.

Lemma

faithfulR

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "astabR", "faithful", "on", "subsetIr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Cayley_repr G

:= actperm <[atrans_acts (atransR G)]>.

Definition

Cayley_repr

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "actperm", "atransR", "atrans_acts" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Cayley_isom G : isom G (Cayley_repr G @* G) (Cayley_repr G).

Proof. exact: faithful_isom (faithfulR G). Qed.

Theorem

Cayley_isom

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Cayley_repr", "faithfulR", "faithful_isom", "isom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Cayley_isog G : G \isog Cayley_repr G @* G.

Proof. exact: isom_isog (Cayley_isom G). Qed.

Theorem

Cayley_isog

finite_group

finite_group/action.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "ssrnotations", "eqtype", "ssrnat", "div", "seq", "prime", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient" ]

[ "Cayley_isom", "Cayley_repr", "isog", "isom_isog" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d